Speakers crackle when I turn on car after replacing head unit

I replaced my 2005 Toyota RAV4 OEM stereo with an aftermarket touchscreen unit. The wires were connected using shrink wrap with built-in solder, and the connections to the harnesses are nice and snug.
The speakers now crackle only when I turn on the car. Any tips to fix this that might avoid having to install new hardware to delay the surge of power to the amp when starting up?
 
Yes, whether or not I turn off the unit before switch off the car I hear a light crackling just as I turn over the car. Then the speakers function fine.
I don’t mind the light crackling, just hope that it won’t really mess up the speakers if I can’t fix it.
 
Plug the OEM head unit back in and confirm that there is no noise.

If there is no noise, measure the DC voltage on all of the head unit output wires that drive the amplifier inputs. Do this with the head unit on and the head unit off and also with the ignition off. Place the black probe on a known good ground (the case of the head unit, preferred).

I don't need to know the precise voltage on every wire. I'm simply needing to know if the voltage is approximately 6v in each test or approximately 0v.